For solving the generalized equation \documentclass[12pt]{minimal}
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\begin{document}$$f(x)+F(x) \ni 0$$\end{document}, where \documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} is a smooth function and \documentclass[12pt]{minimal}
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\begin{document}$$F$$\end{document} is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}-x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$\end{document}where \documentclass[12pt]{minimal}
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\begin{document}$$Df$$\end{document} is the derivative of \documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} and the sequence of mappings \documentclass[12pt]{minimal}
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\begin{document}$$R_k$$\end{document} represents the inexactness. We show how regularity properties of the mappings \documentclass[12pt]{minimal}
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\begin{document}$$f+F$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$R_k$$\end{document} are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.
